Osmosis
Osmosis is the spontaneous net movement of solvent molecules through a selectively permeable membrane into a region of higher solute concentration, in the direction that tends to equalize the solute concentrations on the two sides.
Simply put if we separate any solvent like water with a permeable membrane(it is a type of membrane that allows some particles or ions to pass through , not all) the molecules tends to move to the part that has higher solute(the minor component in a solution for example salt in water) concentration.
Reverse Osmosis
a process by which a solvent passes through a porous membrane in the direction opposite to that for natural osmosis when subjected to a hydrostatic pressure greater than the osmotic pressure.
Lets consider RO water purifiers
Water (not yet purified ) contains lot of impurities or solvents like salts. In case of osmosis if we use some porous membrane(filters) water molecules tends to be collected or stay with the salts. Which is not desirable. Hence we use RO process in purifiers .we try to reverse this natural phenomenon by artificially applying greater pressure then osmotic pressure because of which Water tends to move away from the impurities and we get water free of all the impurities.